Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(896\)\(\medspace = 2^{7} \cdot 7 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.1258815488.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.56.2t1.b.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.1568.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{6} + 5x^{4} - 2x^{2} + 2 \)
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The roots of $f$ are computed in $\Q_{ 71 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 + 38\cdot 71 + 45\cdot 71^{2} + 52\cdot 71^{3} + 50\cdot 71^{4} + 30\cdot 71^{5} +O(71^{6})\)
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| $r_{ 2 }$ | $=$ |
\( 14 + 27\cdot 71 + 58\cdot 71^{2} + 42\cdot 71^{3} + 46\cdot 71^{4} + 34\cdot 71^{5} +O(71^{6})\)
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| $r_{ 3 }$ | $=$ |
\( 27 + 16\cdot 71 + 17\cdot 71^{2} + 67\cdot 71^{3} + 31\cdot 71^{4} + 68\cdot 71^{5} +O(71^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 33 + 67\cdot 71 + 6\cdot 71^{2} + 2\cdot 71^{3} + 5\cdot 71^{4} + 51\cdot 71^{5} +O(71^{6})\)
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| $r_{ 5 }$ | $=$ |
\( 38 + 3\cdot 71 + 64\cdot 71^{2} + 68\cdot 71^{3} + 65\cdot 71^{4} + 19\cdot 71^{5} +O(71^{6})\)
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| $r_{ 6 }$ | $=$ |
\( 44 + 54\cdot 71 + 53\cdot 71^{2} + 3\cdot 71^{3} + 39\cdot 71^{4} + 2\cdot 71^{5} +O(71^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 57 + 43\cdot 71 + 12\cdot 71^{2} + 28\cdot 71^{3} + 24\cdot 71^{4} + 36\cdot 71^{5} +O(71^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 64 + 32\cdot 71 + 25\cdot 71^{2} + 18\cdot 71^{3} + 20\cdot 71^{4} + 40\cdot 71^{5} +O(71^{6})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,7)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $2$ | $8$ | $(1,7,4,3,8,2,5,6)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,3,5,7,8,6,4,2)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.